Q10: What is Feigenbaum's constant?
A10: In a period doubling cascade, such as the logistic equation, consider
the parameter values where period-doubling events occur (e.g.
r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of distances
between consecutive doubling parameter values; let
delta[n] = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to
infinity is Feigenbaum's (delta) constant.
Based on independent computations by Jay Hill and Keith Briggs, it has the
value 4.669201609102990671853... Note: several books have published
incorrect values starting 4.66920166...; the last repeated 6 is a
typographical error.
The interpretation of the delta constant is as you approach chaos, each
periodic region is smaller than the previous by a factor approaching 4.669...
Feigenbaum's constant is important because it is the same for any function
or system that follows the period-doubling route to chaos and has a one-
hump quadratic maximum. For cubic, quartic, etc. there are different
Feigenbaum constants.
Feigenbaum's alpha constant is not as well known; it has the value
2.502907875095. This constant is the scaling factor between x values at
bifurcations. Feigenbaum says, "Asymptotically, the separation of adjacent
elements of period-doubled attractors is reduced by a constant value [alpha]
from one doubling to the next". If d[n] is the algebraic distance between
nearest elements of the attractor cycle of period 2^n, then d[n]/d[n+1]
converges to -alpha.
References:
1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
_Aust. Math. Soc. Gazette_ 16 (1989), p. 89.
2. K. Briggs, A precise calculation of the Feigenbaum constants,
_Mathematics of Computation_ 57 (1991), pp. 435-439.
3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
Mandelsets, _J. Phys._ A24 (1991), pp. 3363-3368.
4. M. Feigenbaum, The Universal Metric Properties of Nonlinear
Transformations, _J. Stat. Phys_ 21 (1979), p. 69.
5. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los
Alamos Sci_ 1 (1980), pp. 1-4. Reprinted in _Universality in Chaos_ ,
compiled by P. Cvitanovic.

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